For the ordinary differential equation

$$\boldsymbol{y}^{\prime}=\boldsymbol{f}(t, \boldsymbol{y}), \quad \boldsymbol{y}(0)=\tilde{\boldsymbol{y}}_{0}, \quad t \geqslant 0$$

where $\boldsymbol{y}(t) \in \mathbb{R}^{N}$ and the function $\boldsymbol{f}: \mathbb{R} \times \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$ is analytic, consider an explicit one-step method described as the mapping

$$\boldsymbol{y}_{n+1}=\boldsymbol{y}_{n}+h \varphi\left(t_{n}, \boldsymbol{y}_{n}, h\right)$$

Here $\varphi: \mathbb{R}_{+} \times \mathbb{R}^{N} \times \mathbb{R}_{+} \rightarrow \mathbb{R}^{N}, n=0,1, \ldots$ and $t_{n}=n h$ with time step $h>0$, producing numerical approximations $\boldsymbol{y}_{n}$ to the exact solution $\boldsymbol{y}\left(t_{n}\right)$ of equation $(*)$, with $\boldsymbol{y}_{0}$ being the initial value of the numerical solution.

(i) Define the local error of a one-step method.

(ii) Let $\|\cdot\|$ be a norm on $\mathbb{R}^{N}$ and suppose that

$$\|\boldsymbol{\varphi}(t, \boldsymbol{u}, h)-\boldsymbol{\varphi}(t, \boldsymbol{v}, h)\| \leqslant L\|\boldsymbol{u}-\boldsymbol{v}\|,$$

for all $h>0, t \in \mathbb{R}, \boldsymbol{u}, \boldsymbol{v} \in \mathbb{R}^{N}$, where $L$ is some positive constant. Let $t^{*}>0$ be given and $\boldsymbol{e}_{0}=\boldsymbol{y}_{0}-\boldsymbol{y}(0)$ denote the initial error (potentially non-zero). Show that if the local error of the one-step method ( $\uparrow$ ) is $\mathcal{O}\left(h^{p+1}\right)$, then

$$\max _{n=0, \ldots,\left\lfloor t^{*} / h\right\rfloor}\left\|\boldsymbol{y}_{n}-\boldsymbol{y}(n h)\right\| \leqslant e^{t^{*} L}\left\|\boldsymbol{e}_{0}\right\|+\mathcal{O}\left(h^{p}\right), \quad h \rightarrow 0$$

(iii) Let $N=1$ and consider equation $(*)$ where $f$ is time-independent satisfying $|f(u)-f(v)| \leqslant K|u-v|$ for all $u, v \in \mathbb{R}$, where $K$ is a positive constant. Consider the one-step method given by

$$y_{n+1}=y_{n}+\frac{1}{4} h\left(k_{1}+3 k_{2}\right), \quad k_{1}=f\left(y_{n}\right), \quad k_{2}=f\left(y_{n}+\frac{2}{3} h k_{1}\right) .$$

Use part (ii) to show that for this method we have that equation (††) holds (with a potentially different constant $L$ ) for $p=2$.